Lesson 2.11
Special Solutions
What happens when the variables disappear? Sometimes math gives you a statement that is always true, or one that is impossible.
Introduction
Usually, we find one value for (e.g., ). But some equations have infinite solutions, and others have none. The clue? The variable terms cancel out completely.
Past Knowledge
You know how to move variables. What if ?
Today's Goal
Identify "Identity" (All Real Numbers) vs. "No Solution" outcomes.
Future Success
Recognizing parallel lines ( vs ) is key for systems of equations.
Key Concepts
True vs. False
When the variable disappears, look at what remains.
TRUE Statement
The equation is an Identity.
FALSE Statement
The equation is a Contradiction.
Symbols to Know
- Infinite Solutions : Any number you plug in works. The left side is identical to the right (e.g., ).
- No Solution : Nothing works. The lines are parallel and never touch.
Worked Examples
Example 1: Identity (True)
BasicSolve for :
Distribute Left Side
Notice they are identical! But let's proceed.
Move Variable
Subtract from both sides.
Example 2: Contradiction (False)
BasicSolve for :
Move Variable
Subtract from both sides.
Example 3: Hidden Identity
AdvancedSolve for :
Distribute Right Side
Move Variable
Subtract .
Common Pitfalls
Writing "0" instead of "No Solution"
If you get , the answer is NOT . That would mean 0 is the value. The answer is "No Solution" (the empty set).
Confusing Identity with Zero
If you get , don't stop and write nothing. It means every number is a solution ().
Real-Life Applications
No Solution: Two parallel train tracks. They will never meet. If you try to calculate their intersection point, you get a "False" statement.
Identity: Two recipes that look different but are actually the same. If and , they are the same line. They touch everywhere.
Practice Quiz
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